SOLUTION: Solve the problem. Please show your work. 1. The population of a colony of bacteria is growing exponentially according to the function below, where t is the time in hours. How l

Algebra ->  Rational-functions -> SOLUTION: Solve the problem. Please show your work. 1. The population of a colony of bacteria is growing exponentially according to the function below, where t is the time in hours. How l      Log On


   

Question 214807: Solve the problem. Please show your work.
1. The population of a colony of bacteria is growing exponentially according to the function below, where t is the time in hours. How long will it take for the population of the colony to grow to 1,000?
B(t)=12*e^(0.2*t)
e=2.718
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
B%28t%29=12%2Ae%5E%28%280.2%2At%29%29
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substitute 1000 for B(t) to get:
1000+=+12%2Ae%5E%28%28.2%2At%29%29
take the natural log of both sides to get:
ln%281000%29+=+ln%2812%2Ae%5E%28.2%2At%29%29
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couple of general properties of logarithms are applicable.
the first is:
ln+%28a%2Ab%29+=+ln%28a%29+%2B+ln%28b%29
the second is:
ln+%28a%5Eb%29+=+b%2Aln%28a%29
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using the first one, we get:
ln%2812%2Ae%5E%28.2%2At%29%29+=+ln%2812%29+%2B+ln%28e%5E%28%28.2%2At%29%29%29
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using the second one:
ln%28e%5E%28.2%2At%29%29+=+.2%2At%2Aln%28e%29
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putting this all together, your equation becomes:
ln%281000%29+=+ln%2812%29+%2B+.2%2At%2Aln%28e%29
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subtract ln%2812%29 from both sides of this equation to get:
ln%281000%29+-+ln%2812%29++=+.2%2At%2Aln%28e%29
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divide both sides of this equation by .2%2Aln%28e%29 to get:
%28ln%281000%29+-+ln%2812%29%29+%2F+%28ln%28e%29%2A.2%29+=+t
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this becomes:
%286.907755279+-+2.48490665%29%2F%281%2A%28.2%29%29
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solve for t to get:
t = 22.11424315 hours.
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the population should increase to 1000 in 22.11424315 hours.
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substitute in original equation of:
B%28t%29=12%2Ae%5E%28%280.2%2At%29%29 to get:
B%28t%29=12%2Ae%5E%28%280.2%2A%2822.11424315%29%29%29 which becomes:
B(t) = 1000 confirming t = 22.11424315 hours is a good value for t.
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Colony will increase to 1000 in 22.11424315 hours.
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